In a bygone analog era, lawmakers and corporate chiefs traveled great distances to swap secrets, to the smoke-filled back rooms of the World Economic Forum in Davos, Switzerland, or the watering holes at the annual Allen & Company conference in Sun Valley, Idaho. But these days, entering the corridors of power is as easy as opening an app.

Secure messaging apps like WhatsApp, Signal and Confide are making inroads among lawmakers, corporate executives and other prominent communicators. Spooked by surveillance and wary of being exposed by hackers, they are switching from phone calls and emails to apps that allow them to send encrypted and self-destructing texts. These apps have obvious benefits, but their use is causing problems in heavily regulated industries, where careful record-keeping is standard procedure.

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Few issues produce bipartisan consensus in Washington these days, but the secure messaging trend has drawn criticism from all sides. Citizens for Responsibility and Ethics in Washington, a watchdog group, recently filed a lawsuit against the Trump administration, alleging that Mr. Trump and his associates were “ignoring or outright flouting” public records laws by using texting apps like Confide. Judicial Watch, a conservative group, sued the Environmental Protection Agency over its staff members’ reported use of Signal, calling the app’s popularity among government workers “disturbing” and saying that it “may make it difficult for their work to be overseen.”